I have another large probability problem, anybody can help?:
Oscar has lost his dog in either forest A (with a priori probability 0.4) or in forest B (with a priori probability 0.6). If the dog is alive and not found by the Nth day of the search, it will die that evening with probability N/(N+2).
If the dog is in A (either dead or alive) and Oscar spends a day searching for it in A, the conditional probability that he will find the dog that day is 0.25. Similarly, if the dog is in B and Oscar spends a day looking for it there, he will find the dog that day with probability 0.15.
The dog cannot go from one forest to the other. Oscar can search only in the daytime, and he can travel from one forest to the other only at night.
All parts of this problem are to be worked separately.
a. In which forest should Oscar look to maximize the probability he finds his dog on the first day of the search?
b. Given that Oscar looked in A on the first day but didn't find his dog, what is the probability that the dog is in A?
c. If Oscar flips a fair coin to determine where to look on the first day and finds the dog on the first day, what is the probability that he looked in A?
d. Oscar has decided to look in A for the first two days. What is the a priori probability that he will find a live dog for the first time on the second day?
e. Oscar has decided to look in A for the first two days. Given the fact that he was unsuccessful on the first day, determine the probability that he does not find a dead dog on the second day.
f. Oscar finally found his dog on the fourth day of the search. He looked in A for the first 3 days and in B on the fourth day. What is the probability he found his dog alive?
g. Oscar finally found his dog late on the fourth day of the search. The only other thing we know is that he looked in A for 2 days and in B for 2 days. What is the probability he found his dog alive?
Quite long isn't it?